theorem Th43:
  still_not-bound_in (P!ll) = Bound_Vars(P!ll)
proof
  set l1 = the_arguments_of (P!ll);
A1: P!ll is atomic by QC_LANG1:def 18;
  then consider
  n being Nat, P9 being (QC-pred_symbol of n,Al), ll9 being
  QC-variable_list of n,Al such that
A2: l1 = ll9 and
A3: P!ll = P9!ll9 by QC_LANG1:def 23;
  Bound_Vars(P!ll) = Bound_Vars(l1) by A1,SUBSTUT1:3;
  then
A4: Bound_Vars(P!ll) = { l1.i : 1 <= i & i <= len l1 & l1.i in
  bound_QC-variables(Al)} by SUBSTUT1:def 7;
  still_not-bound_in (P!ll) = still_not-bound_in ll by QC_LANG3:5;
  then
A5: still_not-bound_in (P!ll) = variables_in(ll,bound_QC-variables(Al)) by
QC_LANG3:2;
A6: (<*P9*>^ll9).1 = P9 & (<*P*>^ll).1 = P by FINSEQ_1:41;
  P!ll = <*P*>^ll & P9!ll9 = <*P9*>^ll9 by QC_LANG1:8;
  then ll9 = ll by A3,A6,FINSEQ_1:33;
  hence thesis by A4,A5,A2,QC_LANG3:def 1;
end;
